The equation of eigen-values in a general Euclidean Schwarzschild metric

TL;DR

This paper generalizes the Euclidean Schwarzschild metric by including a non-zero cosmological constant and replacing the hypersphere with an Einstein variety, deriving eigen-value equations, and analyzing stability.

Contribution

It introduces a more general metric framework and derives the differential equation for Laplace eigen-values, including cases with non-zero cosmological constant and discrete eigen-values.

Findings

Eigen-value differential equation is derived for the generalized metric.

The determinant of the Laplacian is calculated using discrete eigenvalues.

Stability analysis of the Euclidean Schwarzschild metric is performed.

Abstract

A metric more general than the Euclidean Schwarzschild-Tangherlini metric is considered. The cosmological constant is not necessarily Zero, and the hypersphere is replaced by an Einstein variety. A differential equation that derives from the equation of eigen-values of the Laplace operator in this metric is studied. Two cases are considered, (i) the cosmological constant is not Zero, and (ii) the cosmological constant is Zero, and the set of eigen-values is discrete. Infinity is a singular point of this differential equation. In these cases only, it is regular. Series of powers solve the differential equation. With the eigenvalues from the discrete set, the determinant of the Laplacian is calculated. This is equal to the period of the imaginary time coordinate. Finally, the stability of the Euclidean Schwarzschild metric is investigated at the classical level.